Search Results

For any (reduced) cohomology theory $\tilde H^*$, not necessarily ordinary, addition in $\tilde H^r(\Sigma X)$ is induced by the pinch map $\Sigma X \to \Sigma X\ \vee \Sigma X$, using the natural i …

But there is a standard old answer. Take the total singular complex, the geometric realization of the simplicial set whose $n$-simplices are the singular $n$-simplices of a space $X$. That is a per …

The term ``RO(G)-graded spectrum'' is a misnomer, not to be used. It is never used in the literature I'm familiar with, and it never should be. There are several Quillen equivalent models for the cat …

This is a very special case of a very general result. A quasi-isomorphism of DGAs identifies Massey products (not just triple products but matrix Massey products of all sizes). See for example Theore …

There is a strongly related result of Gaunce Lewis that I would like to advertise. We have different kinds of $G$-spectra for different ``universes'' $U$, which are countable sums of representations …

As usual, I apologize for excess concision. Let $A$ be a based CW complex, $X$ and $Y$ well-pointed spaces, $f\colon X\to Y$ a (weak) equivalence. The first claim in 6.9(i) is that $[X\wedge A,Z] \co …

I would like to just comment but don't see how. Christian, here's an answer to your last question. For based spaces, you can just do what you want by hand, as Fernando suggested, taking care to use di …

The proof of Spanier's 6.5.12 starts from a free chain complex with homology of finite type and replaces it by a quasi-isomorphic free chain complex of finite type. Then the conclusion reduces direct …

For any space $F$ you can form the topological monoid $Aut(F)$ and take its classifying space. That will classify Hurewicz fibrations with fiber $F$. A little care is needed since $Aut(F)$ won't gener …

"Basepoints" of "based" $G$-spaces are required to be $G$-fixed. Conceptually, a "basepoint" in an object $X$ of any category with a terminal object wants to be a map from the terminal object into $X …

Yes. This is classical, maybe originally in Milnor and Moore's paper on Hopf algebras. For a recent exposition see for example "More concise algebraic topology" by Kate Ponto and myself. If $X$ is a …

There is a nice theorem in geometric topology that a Serre fibration between actual CW complexes (not just homotopy types thereof) is in fact a Hurewicz fibration. It's in a paper by Steinberger and W …

The phrasing of your question prompts me to emphasize a model categorical difference between spaces and simplicial sets. With the standard weak equivalences, there is just one standard model structur …

I'm feeling mischievous. The history of the Kervaire invariant problem is strewn with false proofs, Jim Milgram's is only one of many. A student of mine (who I will leave nameless) had a preprint (a …

Expressed in terms of the homotopy pullback $N(f,g)$ of a pair of based maps $f\colon X\longrightarrow A$ and $g\colon Y\longrightarrow A$, the long exact sequence is Corollary 2.2.3 of May and Ponto …